×

zbMATH — the first resource for mathematics

Hypoelastic soft tissues. I: Theory. (English) Zbl 1320.74021
Summary: Refinements are made to an existing hypoelastic theory developed by A. D. Freed [“Anisotropy in hypoelastic soft-tissue mechanics. I: Theory, II: Simple extensional experiments. J. Mech. Mater. Struct. 3, No. 5, 911–928 (2008), doi:10.2140/jomms.2008.3.911, ibid. 4, No. 6, 1005–1025 (2009), doi:10.2140/jomms.2009.4.1005] for the purpose of modeling the passive response of soft, fibrous, biological tissues. Oldroyd’s operators [J. G. Oldroyd, Proc. R. Soc. A 200, 523–541 (1950; Zbl 1157.76305)] for convected differentiation and integration, which he derived from the tensor transformation law, are re-derived here from an integral equation defined in the polar configuration. Fields that obey these convected polar operators are said to be viable tensor fields, from which a new definition for strain and its rate are obtained and applied to a hypoelastic theory for tissue. Anisotropy is addressed through a material tensor, from which viable tensor fields describing fiber strain and strain rate are constructed. Material anisotropy and material constitution are handled separately for maximum flexibility. Isochoric hypoelastic models for isotropic, anisotropic, and fiber/matrix composite materials are derived. A material function is introduced to address special attributes that biological fibers impart on tissue behavior, four of which are proposed that represent various ways through which the fiber constituents might be described. Application to in-plane biaxial deformation is the focus of part II of this paper [Acta Mech. 213, No. 1–2, 205–222 (2010; Zbl 1320.74020)].

MSC:
74B20 Nonlinear elasticity
74L15 Biomechanical solid mechanics
74C99 Plastic materials, materials of stress-rate and internal-variable type
74E10 Anisotropy in solid mechanics
PDF BibTeX Cite
Full Text: DOI
References:
[1] Marsden J.E., Hughes T.J.R.: Mathematic Foundations of Elasticity. Prentice-Hall, Englewood Cliffs (1983) (republished by Dover Publications, Mineola, 1994) · Zbl 0545.73031
[2] Ogden R.W.: Non-Linear Elastic Deformations. John Wiley, New York (1984) (republished by Dover Publications, Mineola, 1997) · Zbl 0541.73044
[3] Holzapfel G.A.: Nonlinear Solid Mechanics: a Continuum Approach for Engineering. Wiley, Chichester (2000) · Zbl 0980.74001
[4] Truesdell C.: Hypoelasticity. J. Ration. Mech. Anal. 4, 83–133 (1955)
[5] Criscione J.C., Sacks M.S., Hunter W.C.: Experimentally tractable, pseudo-elastic constitutive law for biomembranes: II. Application. J. Biomech. Eng. 125, 100–105 (2003)
[6] Fung Y.C.: Elasticity of soft tissues in simple elongation. Am. J. Physiol. 28, 1532–1544 (1967)
[7] Humphrey J.D.: Continuum biomechanics of soft biological tissues. Proc. R. Soc. Lond. A-459, 3–46 (2002) · Zbl 1116.74385
[8] Humphrey J.D.: Biological soft tissues. In: Sharpe, W.N.J. (eds) Springer Handbook of Experimental Solid Mechanics, pp. 169–185. Springer, New York (2008)
[9] Sacks M.S.: Biaxial mechanical evaluation of planar biological materials. J. Elast. 61, 199–246 (2000) · Zbl 0972.92004
[10] Sacks M.S., Sun W.: Multiaxial mechanical behavior of biological materials. Annu. Rev. Biomed. Eng. 5, 251–284 (2003)
[11] Viidik A.: Functional properties of collagenous tissues. Int. Rev. Connect. Tissue Res. 6, 127–215 (1973)
[12] Weiss J.A., Gardiner J.C.: Computational modeling of ligament mechanics. Crit. Rev. Biomed. Eng. 29, 303–371 (2001)
[13] Fung Y.C.: Biomechanics: Mechanical Properties of Living Tissues, 2nd edn. Springer, New York (1993)
[14] Humphrey J.D.: Cardiovascular Solid Mechanics; Cells, Tissues, and Organs. Springer, New York (2002)
[15] Freed A.D., Diethelm K.: Caputo derivatives in viscoelasticity: a non-linear finite-deformation theory for tissue. Fractional Calc. Appl. Anal. 10(3), 219–248 (2007) · Zbl 1152.26303
[16] Freed A.D., Einstein D.R., Vesely I.: Invariant formulation for dispersed transverse isotropy in aortic heart valves: an efficient means for modeling fiber splay. Biomech. Model. Mechanobiol. 4, 100–117 (2005)
[17] Greenshields C.J., Weller H.G.: A unified formulation for continuum mechanics applied to fluid-structure interaction in flexible tubes. Int. J. Numer. Methods Eng. 64, 1575–1593 (2005) · Zbl 1122.74379
[18] Freed A.D.: Anisotropy in hypoelastic soft-tissue mechanics, I: theory. J. Mech. Mater. Struct. 3(5), 911–928 (2008)
[19] Freed A.D.: Anisotropy in hypoelastic soft-tissue mechanics, II: simple extensional experiments. J. Mech. Mater. Struct. 4(6), 1005–1025 (2009)
[20] Dienes J.K.: On the analysis of rotation and stress rate in deforming bodies. Acta Mech. 32, 217–232 (1979) · Zbl 0414.73005
[21] Dienes J.K.: A discussion of material rotation and stress rate. Acta Mech. 65, 1–11 (1986) · Zbl 0603.73045
[22] Dienes, J.K.: Finite Deformation of Material with an Ensemble of Defects. Tech. Rep. LA–13994–MS, Los Alamos National Laboratory (2003)
[23] Freed, A.D., Einstein, D.R., Sacks, M.S.: Hypoelastic Soft Tissues, Part II: in-plane biaxial experiments. Acta Mech. (2010) (submitted) · Zbl 1320.74020
[24] Green A.E., Naghdi P.M.: A general theory of an elastic-plastic continuum. Arch. Ration. Mech. Anal. 18, 251–281 (1965) · Zbl 0133.17701
[25] Lodge A.S.: Elastic Liquids: An introductory vector treatment of finite-strain polymer rheology. Academic Press, London (1964)
[26] Lodge A.S.: Body Tensor Fields in Continuum Mechanics: With applications to polymer rheology. Academic Press, New York (1974)
[27] Oldroyd J.G.: On the formulation of rheological equations of state. Proc. R. Soc. Lond. A 200, 523–541 (1950) · Zbl 1157.76305
[28] Noll W.: A mathematical theory of the mechanical behavior of continuous media. Arch. Ration. Mech. Anal. 2, 197–226 (1958) · Zbl 0083.39303
[29] Lodge A.S.: On the use of convected coordinate systems in the mechanics of continuous media. Proc. Camb. Philos. Soc. 47, 575–584 (1951) · Zbl 0042.18702
[30] Meyers A., Schieße P., Bruhns O.T.: Some comments on objective rates of symmetric Eulerian tensors with applications to Eulerian strain rates. Acta Mech. 139, 91–103 (2000) · Zbl 0984.74004
[31] Bird R.B., Armstrong R.C., Hassager O.: Dynamics of Polymeric Liquids. Fluid Mechanics, vol. 1, 2nd edn. Wiley, New York (1987)
[32] Zaremba, S.: Sur une forme perfectionnée de la théorie de la relaxation. Bull. Acad. Cracovie 594–614 (1903)
[33] Jaumann G.: Geschlossenes System physikalischer und chemischer Differentialgesetze. Sitzungsber. d. Kaiserlichen Akad. Wiss. Math. Naturwiss. Kl. 120, 385–530 (1911) · JFM 42.0854.05
[34] Farahani K., Naghdabadi R.: Basis free relations for the conjugate stresses of the strains based on the right stretch tensor. Int. J. Solids Struct. 40, 5887–5900 (2003) · Zbl 1059.74004
[35] Green G.: On the propagation of light in crystallized media. Trans. Camb. Philos. Soc. 7, 121–140 (1841)
[36] Signorini, A.: Sulle deformazioni thermoelastiche finite. In: Oseen, C.W., Weibull, W. (eds.), Proceedings of the 3rd International Congress for Applied Mechanics, vol. 2, Ab. Sveriges Litografiska Tryckerier, Stockholm, pp. 80–89 (1930) · JFM 56.0687.02
[37] Johnson M.W. Jr., Segalman D.: A model for viscoelastic fluid behavior which allows non-affine deformation. J. NonNewton. Fluid Mech. 2, 255–270 (1977) · Zbl 0369.76008
[38] Szabó L., Balla M.: Comparison of some stress rates. Int. J. Solids Struct. 25, 279–297 (1989)
[39] Treloar L.R.G.: The Physics of Rubber Elasticity. 3rd edn. Clarendon Press, Oxford (1975) · Zbl 0347.73042
[40] Fung Y.-C.: Biorheology of soft tissues. Biorheology 10, 139–155 (1973)
[41] Stewart G.W.: Introduction to Matrix Computations. Computer Science and Applied Mathematics. Academic Press, New York (1973) · Zbl 0302.65021
[42] Sokolnikoff I.S.: Tensor Analysis: Theory and applications to geometry and mechanics of continua, 2nd edn., Applied Mathematics Series. Wiley, New York (1964) · Zbl 0121.38204
[43] Sellaro T.L., Hildebrand D., Lu Q., Vyavahare N., Scott M., Sacks M.S.: Effects of collagen fiber orientation on the response of biologically derived soft tissue biomaterials to cyclic loading. J. Biomed. Mater. Res. 80A, 194–205 (2007)
[44] Spencer A.J.M.: Deformations in Fibre-reinforced Materials. Oxford Science Research Papers. Clarendon Press, Oxford (1972) · Zbl 0238.73001
[45] Gasser T.C., Ogden R.W., Holzapfel G.A.: Hyperelastic modelling of arterial layers with distributed collagen fibre orientations. J. R. Soc. Interface 3, 15–35 (2006)
[46] Rajagopal K.R., Tao L.: Mechanics of Mixtures. World Scientific, River Edge (1995) · Zbl 0941.74500
[47] Lanir Y.: Mechanisms of residual stress in soft tissues. J. Biomech. Eng. 131, 044506 (2009)
[48] Lokshin O., Lanir Y.: Viscoelasticity and preconditioning of rat skin under uniaxial stretch: microstructural constitutive characterization. J. Biomech. Eng. 131, 031009 (2009)
[49] Mow V.C., Kuei S.C., Lai W.M., Armstrong C.G.: Biphasic creep and stress relaxation of articular cartilage in compression: theory and experiments. J. Biomech. Eng. 102, 73–83 (1980)
[50] Lai W.M., Hou J.S., Mow V.C.: A triphasic theory for the swelling and deformation behaviors of articular cartilage. J. Biomech. Eng. 113, 245–258 (1991)
[51] Humphrey J.D., Rajagopal K.R.: A constrained mixture model for arterial adaptations to a sustained step change in blood flow. Biomech. Model. Mechanobiol. 2, 109–126 (2003)
[52] Freed A.D., Doehring T.C.: Elastic model for crimped collagen fibrils. J. Biomech. Eng. 127, 587–593 (2005)
[53] Ogden R.W.: Large deformation isotropic elasticity–on the correlation of theory and experiment for incompressible rubberlike solids. Proc. R. Soc. Lond. A-326, 565–584 (1972) · Zbl 0257.73034
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.